Nonlocal operators: Nonlocal operators arise in various areas such as nonlocal and fractional diffusions [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, de-noising and regularization by nonlocal kernels [10]}, [11]}, [12]}, multi-agent systems with nonlocal interaction [13]}, [14]}, [15]} and nonlocal networks [16]}, [17]}.
The inverse problem for nonlocal diffusions has been studied in [18]}, [19]} from a single solution. To discover hidden nonlocal physical laws from data, a parametric nonlocal kernel learning approach has been proposed in [20]}, [21]}, where the coefficients of Bernstein polynomials are learnt with physics-based constraints and a Tikhonov regularization. Beyond the linear nonlocal model and the nonlocal kernel regression methods, nonlocal operators were further combined with neural networks, and nonlocal kernel networks were developed for learning maps between high-dimensional variables in dynamical systems [22]}, [23]} or function spaces [24]}, [25]}. An attractive feature of these nonlocal kernel/operator learning methods is the generalizability among approximations corresponding to different underlying levels of resolution and discretization.
However, as seen in [26]}, [21]}, [24]}, [25]}, neither the nonlocal kernel learning methods nor nonlocal kernel networks yield estimator convergence when trained on finer resolution, and the test error may even increase. This fact indicates the possible ill-posedness of the learning problem. In this work, we tackle this issue by introducing a new regularization method based on a data-adaptive RKHS in a nonparametric learning approach.
